First as to the kits, Lee Kits are a good economical way to get started handloading. Are they the best, no, but very serviceable and a good value for the money.
As to your powder question, there are a few variables that make the answer to that a bit difficult. I would look through your loading manual (I like this one for general reference work such as this) and see what powders you find compatible with the chamberings you wish to load for.
Off the top of my head, for the 222, 250 Sav, and 243 I would look at Varget and/or RL15 and see how they rate with the bullet weights you are wanting to use. I ran some 120gr bullets in a 250 Sav and some 100gr in a 243 and both of those seem to like RL19, but the lighter bullets seemed to favor Varget.
FWIW when I first started out loading I tried to consolidate and keep to one powder for rifle stuff (I was loading 30-06, 22-250, and 308 with plans to move into 30-30 and 223) and I found at that time that Varget was listed as at least decent for all of those so I ran with it. Over time I found it was a good all-around choice working decently for most, but not optimum. I still use it exclusively in my 308, 6BR, and 243AI and for most loads in my 22-250.
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